Average Error: 0.4 → 0.4
Time: 1.3m
Precision: 64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{\frac{-1}{2}}\]
double f(double k, double n) {
        double r8360278 = 1.0;
        double r8360279 = k;
        double r8360280 = sqrt(r8360279);
        double r8360281 = r8360278 / r8360280;
        double r8360282 = 2.0;
        double r8360283 = atan2(1.0, 0.0);
        double r8360284 = r8360282 * r8360283;
        double r8360285 = n;
        double r8360286 = r8360284 * r8360285;
        double r8360287 = r8360278 - r8360279;
        double r8360288 = r8360287 / r8360282;
        double r8360289 = pow(r8360286, r8360288);
        double r8360290 = r8360281 * r8360289;
        return r8360290;
}
↓
double f(double k, double n) {
        double r8360291 = 2.0;
        double r8360292 = atan2(1.0, 0.0);
        double r8360293 = r8360291 * r8360292;
        double r8360294 = n;
        double r8360295 = r8360293 * r8360294;
        double r8360296 = 1.0;
        double r8360297 = k;
        double r8360298 = r8360296 - r8360297;
        double r8360299 = r8360298 / r8360291;
        double r8360300 = pow(r8360295, r8360299);
        double r8360301 = -0.5;
        double r8360302 = pow(r8360297, r8360301);
        double r8360303 = r8360300 * r8360302;
        return r8360303;
}


\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{\frac{-1}{2}}

Error

Bits error versus k

Bits error versus n

Derivation

  1. Initial program 0.4

    \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
  2. Using strategy rm

  3. Applied pow10.4

    \[\leadsto \frac{1}{\sqrt{\color{blue}{{k}^{1}}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  4. Applied sqrt-pow10.4

    \[\leadsto \frac{1}{\color{blue}{{k}^{\left(\frac{1}{2}\right)}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  5. Applied pow-flip0.4

    \[\leadsto \color{blue}{{k}^{\left(-\frac{1}{2}\right)}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  6. Simplified0.4

    \[\leadsto {k}^{\color{blue}{\frac{-1}{2}}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]

  7. Final simplification0.4

    \[\leadsto {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {k}^{\frac{-1}{2}}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  (* (/ 1 (sqrt k)) (pow (* (* 2 PI) n) (/ (- 1 k) 2))))